For a given complex Hermitian matrices $A_1,A_2$ and $A_3$, I have the following problem:
\begin{equation}\tag{1} \begin{array}{ll} \underset{k\geq1,X\in\mathbb{C^{n\times n}}}{\text{minimize}} & k\\ \text{subject to} & \mathrm{trace}((A_1+(k+\frac{1}{k})A_2)X)=0,\\&\mathrm{trace}(A_3X)=0.\end{array} \end{equation}
I want to solve $(1)$ using bisection search in $k$. For a fixed $k$, the problem $(1)$ is convex. I want to first find its dual:
\begin{equation}\tag{2} \begin{array}{ll} \underset{v_1,v_2\in\mathbb{R}}{\text{maximize}} & 0\\ \text{subject to} & v_1(A_1+(k+\frac{1}{k})A_2)+v_2A_3\preceq0.\end{array} \end{equation}
I have the following questions:
- In a research paper, is it possible to write an optimization problem that maximizes a constant?
- If I rewrite the dual problem $(2)$ as the following feasibility problem:
\begin{equation}\tag{3} \begin{array}{ll} \underset{v_1,v_2\in\mathbb{R}}{\text{find}} & v_1,v_2\\ \text{subject to} & v_1(A_1+(k+\frac{1}{k})A_2)+v_2A_3\preceq0,\end{array} \end{equation}
Can I still call the problem $(3)$ a dual problem of $(1)$, or it is considered a bad practice to connect minimization problem and feasibility problem as a dual of each other?